the Pythagorean Theorem

(This lesson also appears in the Algebra I curriculum.
If you studied it there, then just quickly review, and move on to the next section!)

A 90° angle is called a right angle.
A right triangle is a triangle with a 90° angle.

In a right triangle, the side opposite the 90° angle is called the hypotenuse
and the remaining two sides are called the legs.

The angles in any triangle add up to 180°.

In any triangle, the longest side is opposite the largest angle,
and the shortest side is opposite the smallest angle.
Thus, in a right triangle, the hypotenuse is always the longest side.

The Pythagorean Theorem gives a beautiful relationship between the lengths of the sides in a right triangle:
the sum of the squares of the shorter sides is equal to the square of the hypotenuse.
Furthermore, if a triangle has this kind of relationship between the lengths of its sides, then it must be a right triangle!

THE PYTHAGOREAN THEOREM

Let

$\,\,T\,\,$ be a triangle with sides of lengths

$a$,

$b$, and

$c$,
where

$\,c\,$ is the longest side (if there is a longest side). Then,

$T\,$ is a right triangle if and only if

$a^2 + b^2 = c^2\,$.

Click here for a geometric proof of the Pythagorean Theorem

EXAMPLES:

Question: Suppose that two angles in a triangle are 60° and 30°.
Is it a right triangle? Answer YES, NO, or MAYBE.

Solution: Yes.
The third angle must be 180° - 60° - 30° = 90° .

Question: Suppose that a triangle has a 100° angle.
Is it a right triangle? Answer YES, NO, or MAYBE.

Solution: No.
The remaining two angles must sum to 80°, so neither remaining angle is a 90° angle.

Question: Suppose that a triangle has a 70° angle.
Is it a right triangle? Answer YES, NO, or MAYBE.

Solution: Maybe.
The remaining two angles must sum to 110°, so one of the remaining angles could be a 90° angle.

Question: Suppose the legs of a right triangle have lengths $\,3\,$ and $\,x\,$,
and the hypotenuse has length $\,5\,$. Find $\,x\,$.

Solution:

$3^2 + x^2 = 5^2$

$9 + x^2 = 25$

$x^2 = 16$

$x = 4$

Note:

$x\,$ cannot equal

$\,\text{-}4\,$, because lengths are always positive.

The 3-4-5 TRIANGLE is a well-known right triangle.
Multiplying all the sides of a triangle by the same positive number does not change the angles.
Thus, if you multiply the sides of a 3-4-5 triangle by any positive real number k ,
then you will still have a right triangle.
For example, these are all right triangles:

6-8-10	( k = 2 )
9-12-15	( k = 3 )
1.5-2-2.5	( k = 0.5 )
3π-4π-5π	( k = π )

and so on!

Question: Suppose a triangle has sides of lengths

$1\,$,

$\sqrt{3}$, and

$2\,$. Is it a right triangle?

Solution: YES.
Since

$2 > \sqrt{3}\,$, the longest side has length 2.
And,

$1^2 + {(\sqrt{3})}^2 = 1 + 3 = 4 = 2^2\,$.

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
More Terminology for Segments and Angles

(MAX is 35; there are 35 different problem types.)